- Apply the property of absolute value equations: the equation \(|x - 8| = 8\) implies either \(x - 8 = 8\) or \(x - 8 = -8\).
- Solve the first case: \(x - 8 = 8\) results in \(x = 16\).
- Solve the second case: \(x - 8 = -8\) results in \(x = 0\).
- The solution is the set of values \(\boxed{0, 16}\), which corresponds to choice A.
Explanation
- Problem Analysis
The problem asks to find all integers \(x\) that satisfy the absolute value equation \(|x - 8| = 8\). An absolute value expression \(|u|\) represents the distance of the number \(u\) from zero on the number line. Specifically, the definition of the absolute value of a real number \(u\) is \(|u| = u\) if \(u \ge 0\), and \(|u| = -u\) if \(u < 0\). Given the equation \(|x - 8| = 8\), we need to identify the values of \(x\) whose distance from 8 is exactly 8 units.
- Strategy for Solving Absolute Value Equations
The absolute value equation \(|y| = a\), where \(a \ge 0\), is equivalent to the two linear equations \(y = a\) and \(y = -a\). In our case, the expression inside the absolute value is \(y = x - 8\), and the value is \(a = 8\). We will split the equation \(|x - 8| = 8\) into two separate cases to solve for \(x\).
- Solving for x
Case 1: \(x - 8 = 8\). Adding 8 to both sides of this equation, we find \(x = 8 + 8 = 16\).
Case 2: \(x - 8 = -8\). Adding 8 to both sides of this equation, we find \(x = -8 + 8 = 0\).
Both \(0\) and \(16\) are integers, which satisfies the requirement of the problem. Thus, the solutions to the equation \(|x - 8| = 8\) are \(x = 0\) and \(x = 16\).
- Conclusion and Final Answer Selection
Having found the solutions to be \(x = 0\) and \(x = 16\), we evaluate the given choices. Choice A states that the values of \(x\) are a specific set. Our findings match this category. Therefore, the correct choice is A, and the values are 0 and 16. The final answer is \(\boxed{0, 16}\).
Examples
Absolute value equations are essential in many practical scenarios where you are interested in a specific distance or difference, regardless of the direction. For instance, if a thermometer is supposed to read 8 degrees Celsius but has a tolerance of 8 degrees, the range of possible actual temperatures could be calculated using \(|T - 8| = 8\). This means the thermometer could be measuring anywhere from 0 degrees Celsius to 16 degrees Celsius. Understanding how to solve these equations helps in quality control and monitoring systems where deviations from a set point are critical.